Mathquery

Theorem Related To Rings :     Theorem 1 :                    If  R is a commutative ring with unit element and M is an ideal of R , then M is maximal ideal of R iff R / M is a field . Proof :          Since R is a commutative ring with unity , therefore R / M is also a commutative ring with unity . The zero element of the ring R / M is M and the unit element of the coset M +1 where 1 is the unit element of R .              Let the ideal M be maximal . Then to prove that R / M is a field .        Let M + b be any non zero element of    R  / M  . Then M + b ≠ M i.e b doesn't belongs to M . To prove that M + b is inversible .          If (b) is the principal ideal of R generated by b, then M(b) is also an ideal of R . Since b doesn't belong to M , therefore the ideal M is properly contained M +(b) . But M is a maximal ideal of R . Hence we must has M +(b) = R       Since 1∈ R , therefore we must obtain 1 on adding an element of M to an element of (b) . The

Definition and Examples Of Rings :  Definition of Rings :                  A non empty set R is said to be a ring (or an associative ring) if in R there are defined two binary operations called addition and multiplication denoted by  + and * such that for all a,b,c ∈ R .  1. a+b ∈ R (closure axiom for addition)  2. a+b = b+a (commutative axiom for                                                                       addition)  3. a+(b+c) = (a+b)+c   (associative axiom for                                                        addition)  4. There exists an element 0 in R such that               a+0 = 0+a =a for all a∈ R        (Existence of additive identity element in R) 5. For every a in R there exists an element         -a  in R such that a + (-a) = (-a) + a = 0  (Existence of additive inverse element in R) 6. ab ∈ R (closure axiom for multiplication ) 7. a(bc) =( ab)c (Associative axiom for                          multiplication ) 8.  {a(b+c) = ab